A007595 a(n) = C_n / 2 if n is even or ( C_n + C_((n-1)/2) ) / 2 if n is odd, where C = Catalan numbers (A000108).

1, 1, 3, 7, 22, 66, 217, 715, 2438, 8398, 29414, 104006, 371516, 1337220, 4847637, 17678835, 64823110, 238819350, 883634026, 3282060210, 12233141908, 45741281820, 171529836218, 644952073662, 2430973304732, 9183676536076, 34766775829452, 131873975875180

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of necklaces of 2 colors with 2n beads and n-1 black ones. - Wouter Meeussen, Aug 03 2002

Number of rooted planar binary trees up to reflection (trees with n internal nodes, or a total of 2n+1 nodes). - Antti Karttunen, Aug 19 2002

Number of even permutations avoiding 132.

Number of Dyck paths of length 2n having an even number of peaks at even height. Example: a(3)=3 because we have UDUDUD, U(UD)(UD)D and UUUDDD, where U=(1,1), D=(1,-1) and the peaks at even height are shown between parentheses. - Emeric Deutsch, Nov 13 2004

Number of planar trees (A002995) on n edges with one distinguished edge. - David Callan, Oct 08 2005

Assuming offset 0 this is an analog of A275165: pairs of two Catalan nestings with index sum n. - R. J. Mathar, Jul 19 2016

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=1..200
• Peter J. Cameron, Some treelike objects Quart. J. Math. Oxford Ser., Vol. 38, No. 2 (1987), pp. 155-183. Note that line 3 on p. 163 has a typo. - _N. J. A. Sloane_, Apr 18 2014
• Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seq., Vol. 3 (2000), Article 00.1.5.
• Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), Article 16.2.4.
• Andrew Gainer-Dewar, Pólya theory for species with an equivariant group action, arXiv preprint arXiv:1401.6202 [math.CO], 2014.
• Toufik Mansour, Counting occurrences of 132 in an even permutation, arXiv:math/0211205 [math.CO], 2002.
• Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022.

Crossrefs

a(n) = A047996(2*n, n-1) for n >= 1 and a(n) = A072506(n, n-1) for n >= 2.

Occurs in A073201 as rows 0, 2, 4, etc. (with a(0)=1 included).

Cf. also A003444, A007123.

Cf. A000108, A000150, A334843.

Programs

Maple

A007595 := n -> (1/2)*(Cat(n) + (`mod`(n,2)*Cat((n-1)/2))); Cat := n -> binomial(2*n,n)/(n+1);

Mathematica

Table[(Plus@@(EulerPhi[ # ]Binomial[2n/#, (n-1)/# ] &)/@Intersection[Divisors[2n], Divisors[n-1]])/(2n), {n, 2, 32}] (* or *) Table[If[EvenQ[n], CatalanNumber[n]/2, (CatalanNumber[n] + CatalanNumber[(n-1)/2])/2], {n, 24}]
Table[(CatalanNumber[n] + 2^n Binomial[1/2, (n + 1)/2] Sin[Pi n/2])/2, {n, 1, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)
Table[If[EvenQ[n],CatalanNumber[n]/2,(CatalanNumber[n]+CatalanNumber[(n-1)/2])/2],{n,30}] (* Harvey P. Dale, Sep 06 2021 *)

PARI

catalan(n) = binomial(2*n, n)/(n+1);
a(n) = if (n % 2, (catalan(n) + catalan((n-1)/2))/2, catalan(n)/2); \\ Michel Marcus, Jan 23 2016

Formula

G.f.: (2-2*x-sqrt(1-4*x)-sqrt(1-4*x^2))/x/4. - Vladeta Jovovic, Sep 26 2003

D-finite with recurrence: n*(n+1)*a(n) -6*n*(n-1)*a(n-1) +4*(2*n^2-10*n+9)*a(n-2) +8*(n^2+n-9)*a(n-3) -48*(n-3)*(n-4)*a(n-4) +32*(2*n-9)*(n-5)*a(n-5)=0. - R. J. Mathar, Jun 03 2014, adapted to offset Feb 20 2020

a(n) ~ 4^n /(2*sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Jul 19 2016

a(2n) = A000150(2n). - R. J. Mathar, Jul 19 2016

a(n) = (A000108(n) + 2^n * binomial(1/2, (n+1)/2) * sin(Pi*n/2))/2. - Vladimir Reshetnikov, Oct 03 2016

Sum_{n>=1} a(n)/4^n = (3-sqrt(3))/2 (A334843). - Amiram Eldar, Mar 20 2022

Extensions

Description corrected by Reiner Martin and Wouter Meeussen, Aug 04 2002